Where does the graph of y = tan theta intersects the graph of y = -1.
tan 45 degrees = 1, now worry about the quadrant (x and y different signs)
You are asking when tan theta = -1
that is at 90+45 degrees
and at 270 + 45 degrees
To find the intersection points between the graphs of y = tan(theta) and y = -1, we need to solve the equation tan(theta) = -1.
Step 1: Rewrite tan(theta) = -1 as sin(theta)/cos(theta) = -1.
Step 2: Multiply both sides of the equation by cos(theta) to eliminate the fraction: sin(theta) = -cos(theta).
Step 3: Divide both sides of the equation by cos(theta) to isolate sin(theta): sin(theta)/cos(theta) = -1.
Step 4: Recall the trigonometric identity tan(theta) = sin(theta)/cos(theta). This allows us to rewrite the equation as tan(theta) = -1.
Step 5: Determine the values of theta that satisfy the equation tan(theta) = -1. These values correspond to the angles for which the tangent function is equal to -1.
The tangent function has a period of pi (180 degrees). Therefore, we can start by finding one solution in the interval theta = [0, pi].
Step 6: Identify the solution in the interval [0, pi]. There is a well-known angle for which tan(theta) = -1 in this interval, which is theta = -pi/4 or -45 degrees.
Step 7: Determine if there are any additional solutions. Since the tangent function has a period of pi, we can find additional solutions by adding multiples of pi to the initial solution.
Adding pi to -pi/4 gives us theta = pi - pi/4 = 3pi/4 or 135 degrees.
Therefore, the graph of y = tan(theta) intersects the graph of y = -1 at two points: theta = -pi/4 and theta = 3pi/4, corresponding to the angles -45 degrees and 135 degrees, respectively.